Quadratic Functions & Complex Numbers

The Imaginary Unit

Defining i and simplifying square roots of negative numbers.

Advanced20 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Inventing a Number to Fill a Gap

Recall from Algebra I: when a quadratic's discriminant is negative, the quadratic formula asks you to take the square root of a negative number — something no real number can do, since a real number squared is never negative. Before reading on, think about this: has mathematics ever run into a wall like this before, where the numbers on hand simply couldn't answer a question the algebra itself was asking?

It has, repeatedly. Whole numbers alone can't answer 3 − 5, so negative numbers were invented. Integers alone can't answer 1 ÷ 3 exactly, so fractions were invented. Rational numbers alone can't answer 'what number squared is 2,' so irrational numbers were invented. Each time, mathematicians didn't discard the question — they extended the number system just enough to answer it, while keeping every old algebra rule intact. The square root of a negative number is the next gap in that same pattern.

Definition — The Imaginary Unit

i is defined as a number satisfying i² = −1 — equivalently, i = √−1. A number of the form bi (b a real number) is called imaginary; a number of the form a + bi is called complex, combining a real part a and an imaginary part b.

Once i is defined, the square root of any negative number simplifies using it: √−9 = √9 · √−1 = 3i. This lets the quadratic formula produce an actual answer even when the discriminant is negative — the 'no real solutions' case from Algebra I is really 'no real solutions, but two complex ones,' a distinction that will matter throughout this unit.

Worked Example — Simplifying a Square Root of a Negative Number

Simplify √−50. Factor out −1: √−50 = √50 · √−1 = √50 · i. Since √50 = √25 · √2 = 5√2, the result is 5√2 · i, usually written 5i√2.

Worked Example — Finding a Pattern in Powers of i

Evaluate i¹ through i⁸. i¹ = i. i² = −1 (by definition). i³ = i² · i = −i. i⁴ = i² · i² = (−1)(−1) = 1. i⁵ = i⁴ · i = i, and the pattern repeats: i, −1, −i, 1, i, −1, −i, 1 — a cycle of length 4, since once i⁴ = 1, multiplying by i again just restarts the same four values.

Complex Plane

Unit circleVectorsAngle arcsPoint mode (click to add a number)

Show z₁

Color

3 + 4i = 5∠53.1301°

Show z₂

Color

-2

1 - 2i = 2.2361∠-63.4349°

Operations

Operation

z₁ + z₂ =

4 + 2i

Step-by-step

z₁ + z₂ = (3 + 1) + (4 + -2)i
= 4 + 2i

Forms — z₁

Rectangular: $z₁ = 3 + 4i$
Polar: z₁ = 5∠53.1301°
Trigonometric: z₁ = 5(cos 53.1301° + i sin 53.1301°)
Exponential (Euler): $z₁ = 5e^{0.9273i}$

Tip

To evaluate any power of i quickly, divide the exponent by 4 and use only the remainder — i²³ behaves exactly like i³ (since 23 = 4×5 + 3), because every complete group of 4 multiplies back to exactly 1 and contributes nothing new.

Common Mistakes

Treating √−9 · √−4 the same as √(−9 · −4) = √36 = 6, applying the real-number radical multiplication rule to negative radicands.
The rule √a · √b = √(ab) only holds when a and b aren't both negative — simplify each square root to a multiple of i first, then multiply: √−9 · √−4 = (3i)(2i) = 6i² = −6, a different (and correct) answer.
Forgetting that i² simplifies to −1 partway through a calculation, leaving i² sitting unevaluated in a final answer.
Every time i² appears in a computation, replace it with −1 immediately — it's a definition, not an approximation, so it should always be substituted.

Key Takeaways

i is defined by i² = −1, extending the real numbers to answer questions like √−1 that reals alone can't.
A complex number a + bi combines a real part and an imaginary part.
Powers of i cycle through i, −1, −i, 1 every four steps, since i⁴ = 1.

Summary

The imaginary unit fills exactly the gap the quadratic formula exposed when discriminants go negative. The next lesson extends ordinary arithmetic — addition, multiplication, division — to these new complex numbers.

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